EVENT DETAILS AND ABSTRACT

Title: Traces in indexed monoidal categories
Speaker: Mike Shulman
Speaker Info: UCSD
Brief Description:
Special Note:
Abstract:

The classical fixed-point index can be realized category-theoretically as a trace in the stable homotopy category, making the Lefschetz fixed-point theorem a simple consequence of functoriality. More refined fixed-point invariants, such as the Reidemeister trace (which supports a converse of the Lefschetz theorem), do not fit into this framework exactly. However, Kate Ponto showed that the Reidemeister trace is actually a trace in a certain bicategory.

Classically, it is obvious that the Reidemeister trace refines the fixed-point index, but it is not as clear how to compare the two abstract points of view. The bridge is an "indexed monoidal category", which gives rise to both traces in an abstract context. This enables an easy comparison theorem, which specializes to the classical fact that the Reidemeister trace factors the fixed-point index. The abstract structure thus provides a natural way to generalize the theory to fiberwise and equivariant situations.

(This is joint work with Kate Ponto.)